Question: If $8 \tan \theta = 3 \cos \theta$ and $0 < \theta < \pi,$ then determine the value of $\sin \theta.$
Solution: We can write the given equation as
\[\frac{8 \sin \theta}{\cos \theta} = 3 \cos \theta.\]Then $8 \sin \theta = 3 \cos^2 \theta.$  Since $\cos^2 \theta = 1 - \sin^2 \theta,$
\[8 \sin \theta = 3 - 3 \sin^2 \theta.\]Then $3 \sin^2 \theta + 8 \sin \theta - 3 = 0,$ which factors as $(3 \sin \theta - 1)(\sin \theta + 3) = 0.$  Since $-1 \le \sin \theta \le 1,$ we must have $\sin \theta = \boxed{\frac{1}{3}}.$